Theorem grpsubfvalg 12748

Description: Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.)

Hypotheses

Ref	Expression
grpsubval.b	|- B = ( Base ` G )
grpsubval.p	|- .+ = ( +g ` G )
grpsubval.i	|- I = ( invg ` G )
grpsubval.m	|- .- = ( -g ` G )

Assertion

Ref	Expression
grpsubfvalg	|- ( G e. V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )

Distinct variable groups:   x, y, B   x, G, y   x, I, y   x, .+ , y

Allowed substitution hints:   .- ( x, y)   V( x, y)

Proof of Theorem grpsubfvalg

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpsubval.m	. 2 |- .- = ( -g ` G )
2		df-sbg 12713	. . 3 |- -g = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) )
3		fveq2 5496	. . . . 5 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		grpsubval.b	. . . . 5 |- B = ( Base ` G )
5	3, 4	eqtr4di 2221	. . . 4 |- ( g = G -> ( Base ` g ) = B )
6		fveq2 5496	. . . . . 6 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
7		grpsubval.p	. . . . . 6 |- .+ = ( +g ` G )
8	6, 7	eqtr4di 2221	. . . . 5 |- ( g = G -> ( +g ` g ) = .+ )
9		eqidd 2171	. . . . 5 |- ( g = G -> x = x )
10		fveq2 5496	. . . . . . 7 |- ( g = G -> ( invg ` g ) = ( invg ` G ) )
11		grpsubval.i	. . . . . . 7 |- I = ( invg ` G )
12	10, 11	eqtr4di 2221	. . . . . 6 |- ( g = G -> ( invg ` g ) = I )
13	12	fveq1d 5498	. . . . 5 |- ( g = G -> ( ( invg ` g ) ` y ) = ( I ` y ) )
14	8, 9, 13	oveq123d 5874	. . . 4 |- ( g = G -> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) = ( x .+ ( I ` y ) ) )
15	5, 5, 14	mpoeq123dv 5915	. . 3 |- ( g = G -> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
16		elex 2741	. . 3 |- ( G e. V -> G e. _V )
17		basfn 12473	. . . . . 6 |- Base Fn _V
18		funfvex 5513	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
19	18	funfni 5298	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
20	17, 16, 19	sylancr 412	. . . . 5 |- ( G e. V -> ( Base ` G ) e. _V )
21	4, 20	eqeltrid 2257	. . . 4 |- ( G e. V -> B e. _V )
22		mpoexga 6191	. . . 4 |- ( ( B e. _V /\ B e. _V ) -> ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) e. _V )
23	21, 21, 22	syl2anc 409	. . 3 |- ( G e. V -> ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) e. _V )
24	2, 15, 16, 23	fvmptd3 5589	. 2 |- ( G e. V -> ( -g ` G ) = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
25	1, 24	eqtrid 2215	1 |- ( G e. V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   _Vcvv 2730   Fn wfn 5193   ` cfv 5198  (class class class)co 5853   e. cmpo 5855   Basecbs 12416   +g cplusg 12480   invgcminusg 12709   -gcsg 12710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-inn 8879  df-ndx 12419  df-slot 12420  df-base 12422  df-sbg 12713

This theorem is referenced by:  grpsubval  12749